Table of Contents - 20.16.6. Extended real and complex numbers, real and complex projective lines
In this section, we indroduce several supersets of the set of real
numbers and the set of complex numbers.
Once they are given their usual topologies, which are locally compact, both
topological spaces have a one-point compactification. They are denoted by
and respectively, defined in df-bj-cchat and
df-bj-rrhat, and the point at infinity is denoted by , defined
in df-bj-infty.
Both and also have "directional compactifications", denoted
respectively by , defined in df-bj-rrbar (already defined as
, see df-xr) and , defined in df-bj-ccbar.
Since does not seem to be standard, we describe it in some detail.
It is obtained by adding to a "point at infinity at the end of each
ray with origin at 0". Although is not an important object in
itself, the motivation for introducing it is to provide a common superset to
both and and to define algebraic operations (addition,
opposite, multiplication, inverse) as widely as reasonably possible.
Mathematically, is the quotient of
by the diagonal multiplicative action of (think of the closed
"northern hemisphere" in ^3 identified with , that
each open ray from 0 included in the closed northern half-space intersects
exactly once).
Since in set.mm, we want to have a genuine inclusion , we
instead define as the (disjoint) union of with a circle at
infinity denoted by . To have a genuine inclusion
, we define and as certain points in
.
Thanks to this framework, one has the genuine inclusions
and and similarly
and .
Furthermore, one has as well as and
.
Furthermore, we define the main algebraic operations on
, which is not very mathematical, but "overloads" the
operations, so that one can use the same notation in all cases.